Stern Gerlach and decoherence

[naima]

In this paper Caldeira gives a model for the Stern Gerlach device in the vacuum.
the incoming particle is described by the tensor product of a space term and a spin term a |u> + b |d>
the SG is in the vacuum (no air around). Under the effect of the spatial variation of the magnetic field, there is entanglement of these two degrees of freedom. the spatial part splits into two Gaussians that deviate and then hardly overlap.
By a partial trace on the external degree of freedom (the spatial position of the Gaussian), one gets a decohered reduced density matrix.
Calfeira writes at the end that if we recombine the paths of these two Gaussians we retrieve the starting state and thus the lost coherences.
Suppose that we have three regions:
In the first Gaussian separate. in the second a device blocks their remoteness and the paths are parallel.
In the third region a device recombines yhe two possible paths.
If in region 2 we have the vacuum we get what Caldeira described with a final recoherence .
Now suppose now that in the second region there are several air molecules (say 1 on the way up and 1 on the way down),I guess that in the Hamiltonian an interaction term must be added
The device in region 3 remaining unchanged, how does QM predict different behaviors for the particle out of region 3 (with or without gas in region 2)?
I recall that in both cases the output particle of region 1 has null non-diagonal terms in the density matrix.

 

[Strilanc]

Any path-dependent interactions will prevent the paths' contributions from interfering/re-cohering later on. For example, that's how the Elitzur-Vaidman bomb tester works.

 

[naima]

Thanks but this is a principle. I am looking for a model in this case.
I am not asking why but how.

 

[Jilang]

So you have a Gaussian interacting with a couple more Gaussians. The resultant wave function would be rather messed up. What would be the chances of it recombining with a similarly messed up wave function to produce a final Gaussian?

 

[naima]

It depends on the number of air molecules. If it is macroscopic it will not be the same as with 2 molecules. I would like to study how it depends of this number. with few molecules the final state could have still much recoherence out of region 3.

 

[Jilang]

Partially blurred then. A slightly broken down Gaussian?

 

[naima]

I think that we should start with an hamiltonian and see later if gaussians are still solutions. here the interaction hamiltonian between spin and spatial position is very simple it is ##\sigma z##! (it is an operator)
Caldeira and Leggett wrote a model for the interaction between a particle and an infinite bath of harmonic oscillators. for each oscillator the hamiltonian is ##C_k X q_k##
Maybe the hamiltonian would be here ##\sigma z + C z q## or something like that?

 

[Strilanc]

With qubits this situation is relatively simple. The interaction is a controlled operation, and its strength is related to the amount of rotation that operation causes (with a maximum at 180 degrees).

Here's a circuit diagram showing what happens as the amount of controlled rotation is varied between two Hadamard gates:

When the rotation is small (yellow spinner on X^t is near the right), the top qubit is staying coherent and most of the amplitude ends up back in the 00 state due to destructive interference. When the rotation is large (yellow spinner on X^t is near the left), the two qubits end up entangled and evenly split between the 00, 01, 10, and 11 states (and their individual marginal states are maximally mixed).

The gaussian case will be more involved, and I don't have the physics knowledge to compute it, but it should display those same basic features. As the interaction strength increases, destructive interference goes away.

 

[naima]

Thanks,
Here in region 2 i am not interested in the gaussian spatial shape. My problem is to know if i will get something different for the spin with or without an additional gas molecule.
What is the matrix model of your gate? Can it be used in this case to describe The interaction?
It is obvious that when we increase the number of air molecules ti becomes harder to get recoherence. As you propose to associate a logical gate to a molecule, is there a way (serial or, parallel) to associate them?

 

[naima]

I got no answer to my question in the first post:
When the particle quits region 1 (the SG) its density matrix is diagonal.
When it enters region 3 (the beams merger) it is diagonal.
If there was gas or no gas in region 2, how does QM explain that a same density matrix for the particle and a same setup give different recoherence possibilities?

 

[naima]

There is a possible solution to my problem. I only considered the density matrix of the internal degree of freedom. Is it possible that the path information is differently encoded in the shape of the wave? Gaussian (with no gas) or more complicated with particles in zone 2? So the beam merger could not erase some path information in the gas case...

Edit. I found the answer. Without gaz the up and down spins are entangled with displaced gaussians. When the merger recombines them there is factorisation with the same gaussian. With gas the situation is no more symmetric.